The chromatic number of signed graphs with bounded maximum average degree

Abstract

A signed graph is a simple graph with two types of edges: positive and negative edges. Switching a vertex v of a signed graph corresponds to changing the type of each edge incident to v.A homomorphism from a signed graph G to another signed graph H is a mapping φ:V(G)→V(H) such that, after switching some of the vertices of G, φ maps every edge of G to an edge of H of the same type. The chromatic number χs(G) of a signed graph G is the order of a smallest signed graph H such that there is a homomorphism from G to H.The maximum average degree mad(G) of a graph G is the maximum of the average degrees of all the subgraphs of G. We denote Mk the class of signed graphs with maximum average degree less than k and Pg the class of planar signed graphs of girth at least g.We prove: •χs(P7)≤5,•χs(M175)≤10 which implies χs(P5)≤10,•χs(M4−8q+3)≤q+1 with q a prime power congruent to 1 modulo 4.